Overview of Tomorrow's Matches in CAF Women's Champions League Group B
The CAF Women's Champions League Group B is set to witness some thrilling matches tomorrow, featuring teams that have been showing exceptional form throughout the tournament. This section will delve into the details of each match, providing expert betting predictions and insights into what to expect from these exciting encounters.
Match 1: Team A vs. Team B
Team A and Team B are set to clash in what promises to be a closely contested match. Both teams have shown strong defensive capabilities, but their attacking prowess could tip the balance. Team A has been known for its fast-paced counter-attacks, while Team B excels in maintaining possession and controlling the tempo of the game.
Betting Predictions
- Team A Win: With a 35% chance, Team A's aggressive style could catch Team B off guard.
- Draw: At 30%, a draw seems likely given both teams' defensive strengths.
- Team B Win:: With a 35% chance, Team B's control over possession might lead them to victory.
Match 2: Team C vs. Team D
This match features two of the most attacking teams in Group B. Team C has been prolific in front of goal, while Team D has demonstrated resilience by often coming back from behind. The clash between these two teams is expected to be high-scoring and full of tactical intrigue.
Betting Predictions
- Over 2.5 Goals:: With an 80% probability, fans can expect plenty of goals.
- Team C Win:: At 40%, their attacking prowess might give them the edge.
- Team D Win:: At a close second with a 35% chance, their comeback ability could see them through.
Tactical Analysis: Key Players and Strategies
In this section, we'll explore the key players who could make a significant impact on tomorrow's matches and analyze the strategies that coaches might employ to secure victory.
Key Players to Watch
- Sarah Johnson (Team A):: Known for her incredible speed and dribbling skills, she could be crucial in breaking down defenses.
- Laura Smith (Team B):: As a playmaker with excellent vision, she can orchestrate attacks and create scoring opportunities for her team.
- Mia Thompson (Team C):: Her striking ability makes her one of the top scorers in the league this season.
- Nina Rodriguez (Team D):: Her tenacity and work rate make her indispensable in midfield battles.
Tactical Approaches
The coaches will need to devise strategies that not only leverage their team's strengths but also exploit their opponent's weaknesses. Here are some potential tactics:
- High Pressing:: Teams like Team C may opt for high pressing to disrupt opponents' build-up play and regain possession quickly.
- Possession-Based Play:: Teams such as Team B might focus on maintaining possession to control the game's tempo and reduce pressure on their defense.
- Cunning Counter-Attacks:: Teams like Team A could rely on swift counter-attacks to catch opponents off guard when they push forward aggressively.
- Balanced Defense-Midfield Transition:: For teams like Team D, ensuring smooth transitions between defense and attack will be key to maintaining stability while exploiting scoring chances when they arise.
- Focused Set-Piece Execution:: Given that many goals come from set-pieces in women’s football tournaments, teams will pay special attention to optimizing their routines during corners and free-kicks.
Detailed Match Previews: What You Need To Know Before Tomorrow’s Games
Match Preview: Team A vs. Team B
This matchup promises an intriguing battle between two tactically sound sides with contrasting styles. While both squads possess strong defensive units capable of stifling opposition attacks effectively throughout games’ duration – it’s likely that whichever team manages first breakthrough will hold significant advantage going forward into subsequent phases post halftime break interval!
A critical factor influencing outcomes here would involve how well either side adapts mid-match if initial plans don’t yield desired results early enough; hence adaptability becomes vital here as well!
Key Factors Influencing Match Outcome:
- The ability of Sarah Johnson from Team A breaking through Laura Smith’s midfield coverage; max_num_values:
[21]: log_fn(prefix + ' ...too many tensors (%d), skipping printout...' % num_tensors)
[22]: max_name_len = max([len(name) for name in tensor_dict.keys()])
shape_str = '(' + ','.join([str(d) for d in tensor_dict[name].shape]) + ')'
if len(shape_str) > max_name_len:
shape_str = shape_str[:max_name_len] + '...'
log_fn(prefix + ' {:>{}} : {}'.format(name.ljust(max_name_len), max_name_len, shape_str))
if tensor_dict[name].is_floating_point():
mean_val = tensor_dict[name].mean().item()
std_val = tensor_dict[name].std().item()
min_val = tensor_dict[name].min().item()
max_val = tensor_dict[name].max().item()
if num_tensors <= max_num_values:
try:
values_strs = ['{:.4g}'.format(v.item()) for v in tensor_dict[name][:max_num_values]]
values_strs += ['...'] if len(tensor_dict[name]) > max_num_values else []
log_fn(prefix + ' Values: [{}, {}, mean={:.4g} std={:.4g}]'.format(min_val,
max_val,
mean_val,
std_val))
except RuntimeError as e:
print('WARNING! Could not print out values because: {}'.format(e))
else:
log_fn(prefix + ' Values : [{}, {}]'.format(', '.join(values_strs[:10]),
', '.join(values_strs[-10:]) if len(values_strs) >10 else ''))
elif tensor_dict[name].dtype == torch.bool:
try:
unique_vals_counts_tuples_list = [(int(k), v.item()) for k,v in tensor_dict[name].unique(return_counts=True)]
unique_vals_counts_tuples_list.sort(key=lambda x:x[-1], reverse=True)
counts_summed_so_far=0
values_to_log=[]
total_summed_count=0
while counts_summed_so_far20:
val_to_log=unique_vals_counts_tuples_list.pop(0)
values_to_log.append(val_to_log)
counts_summed_so_far+=val_to_log[-1]
total_summed_count+=val_to_log[-1]
except RuntimeError as e:
print('WARNING! Could not print out bool values because: {}'.format(e))
else:
if len(unique_vals_counts_tuples_list)==0 or total_summed_count>=tensor_dict[name].numel()//5*4+1:
else:
try:
min_value,max_value=_get_max_min_value(tensor_dict[name])
mean,std=_get_mean_std(tensor_dict[name])
unique_vals_counts_tuples_list=[(int(k),v.item()) for k,v in tensor_dict[name].unique(return_counts=True)]
unique_vals_counts_tuples_list.sort(key=lambda x:x[-1],reverse=True)
counts_summed_so_far=0
values_to_log=[]
total_summed_count=0
while counts_summed_so_far20:
val_to_log=unique_vals_counts_tuples_list.pop(0)
values_to_log.append(val_to_log)
counts_summed_so_far+=val_to_log[-1]
total_summed_count+=val_to_log[-1]
except RuntimeError as e:
else:
if len(unique_vals_counts_tuples_list)==0 or total_summed_count>=tensor_dict[name].numel()//5*4+1:
else:
shape_str='({})'.format(','.join([str(d) for d in tensor.shape]))
if len(shape_str)>max_name_len:
shape_str=shape_str[:max_name_len]+'...'
log_fn(prefix+' {:>{}} : {}'.format(name.ljust(max_name_len),max_name_len,shape_str))
if tensor.is_floating_point():
mean_val=tensor.mean().item()
std_val=tensor.std().item()
min_val=tensor.min().item()
max_val=tensor.max().item()
if num_tensors<=max_num_values:
try:
values_strs=['{:.4g}'.format(v.item()) for v in tensor[:max_num_values]]
values_strs+=['...']if len(tensor)>max_num_values else []
log_fn(prefix+' Values: [{}, {}, mean={:.4g} std={:.4g}]'.format(min_val,max_val,mean_val,std_val))
except RuntimeError as e:
print('WARNING! Could not print out float values because: {}'.format(e))
else:
log_fn(prefix+' Values : [{}, {}]'.format(','.join(values_strs[:10]),','.join(values_strs[-10:])if len(values_strs)>10else''))
elif tensor.dtype==torch.bool:
try:
unique_vals_counts_tuples_list=[(int(k),v.item())for k,vin tensor.unique(return_counts=True)]
unique_vals_counts_tuples_list.sort(key=lambda x:x[-1],reverse=True)
counts_summed_so_far=0
values_to_log=[]
total_summed_count=0
while counts_summed_so_far20:
val_to_log=unique_vals_counts_tuples_list.pop(0)
values_to_log.append(val_to_log)
counts_summed_so_far+=val_to_log[-1]
total_summed_count+=val_to_log[-1]
except RuntimeError as e:
else:
if len(unique_vals_counts_tuples_list)==0ortotal_summed_count>=tensor.numel()//5*4+1:
else:
try:
min_value,max_value=_get_max_min_value(tensor)
mean,std=_get_mean_std(tensor)
unique_vals_counts_tuples_list=[(int(k),v.item())for k,vin tensor.unique(return_counts=True)]
unique_vals_counts_tuples_list.sort(key=lambda x:x[-1],reverse=True)
counts_summed_so_far=0
values_to_log=[]
total_summed_count=0
while counts_summed_so_far20:
val_to_log=unique_vals_counts_tuples_list.pop(0)
values_to_log.append(val_to_log)
counts_summed_so_far+=val_to_log[-1]
total_summed_count+=val.to.log[-1]
except RuntimeError as e:
else:
if len(unique_vals_counts_tuplearXiv identifier: math/0503219
# Automorphism groups of $C^*$-algebras arising from graph $C^*$-algebras
Authors: Hyeoncheol Jeong (Seoul National University), Taejin Choi (Seoul National University)
Date: 23 May 2007
Categories: math.OA math.FA math.RT math.RA math.SP q-alg quant-ph stat.AP stat.TH
## Abstract
We study automorphism groups $Aut(A)$ ($A$ is a $C^*$-algebra) induced by graph $C^*$-algebras associated with row-finite graphs without sinks which are not trees but satisfy certain conditions called "property $(ast)$" (see Definition~ref{defn_property_star}). We show that $Aut(A)$ is always countable unless $A$ is AF-algebra or UHF-algebra.
## Introduction.
The study on automorphisms of $C^*$-algebras was initiated by Kadison [Kad60] who showed that every automorphism group $mathrm{Aut}(B(H))$ ($H$ is an infinite dimensional Hilbert space) is uncountable by using some basic facts about operator algebras such as Gelfand-Naimark-Segal representation theorem ([Bla98]) or Wigner’s theorem ([Wig31]). In particular he proved that there exist uncountably many unitary operators which cannot be connected by continuous paths.
In [Dix89], Dixmier classified all possible automorphism groups $mathrm{Aut}(M_n(mathbb{C}))$ where $M_n(mathbb{C})$ denotes an algebra consisting all complex square matrices whose entries are complex numbers with size $n$. He showed that $mathrm{Aut}(M_n(mathbb{C}))$ coincides with projective unitary group $mathcal{PU}_n$, i.e., quotient group obtained by dividing unitary group $mathcal{U}_n={alphain M_n(mathbb{C})|alpha^*alpha=alphaalpha^*=I_n}$ by its center consisting scalar multiples of identity matrix $I_n$. Note that $mathcal{PU}_nsimeq O_{n^2}/O_1simeq O_{n^2-1}$ where $O_m$ denotes orthogonal group consisting orthogonal matrices whose entries are real numbers with size $m$. He also showed that every subgroup of $mathrm{Aut}(M_n(mathbb{C}))$ coincides with some subgroup $mathcal{PU}_k$, i.e., every automorphism group induced by finite dimensional matrix algebras must be finite dimensional Lie group.
In [KL97], Kirchberg-Longo studied automorphisms induced by UHF-algebras which were introduced by Powers [Pow57] around middle time period of last century. They proved that any automorphism group induced by UHF-algebra is countable.
In [Jeo04a], Jeong studied automorphisms induced by graph $C^*$-algebras which were introduced independently by Kumjian-Pask [KP92] around middle time period of last century and Renault [Ren81] earlier time period around beginning time period of last century though Renault considered more general cases including non-symmetric cases than Kumjian-Pask did.
Let us recall briefly about graph $C^*$-algebras before stating main results obtained here.
For any directed graph ${E}=(E^o,E^e,r,s)$ ($E^o$ denotes vertices set consisting zero or more elements; $E^e$ denotes edges set; range map/range function/edge range map/range function denoted by “r” associates each edge with its terminal vertex; source map/source function/edge source map/source function denoted by “s” associates each edge with its initial vertex; all maps are assumed injective unless otherwise stated explicitly.) we define *path space* ${E}^{path}$ which consists all finite paths having length at least one such that any consecutive edges share common vertex i.e.,
[ {E}^{path}:={mu=e_{{}_{i_{{}_{|{mu}|}}}}cdots e_{{}_{i_{{}_{{}_{{}_{phantom{|}{}}}}}{}_{i_{{}_{|{mu}|}-{}_{{}_{{}_{{}_{phantom{|}{}}}}}{}_{_}}}}} | forall j {rm s.t}. |{mu}|>|j|+{}_{{}_{{}_{{}_{phantom{|}{}}}}}{}_{_}, r(e_j)=s(e_{j+{}_{{}_{{}_{{}_{phantom{|}{}}}}}{}_{_}})}. ]
Here ${}_{_}$ denotes empty string which plays role analogous role played by zero element playing role within additive monoid structure under concatenation operation over strings whereas “${}phantom{|}{}$” denotes empty path which plays analogous role played by identity element playing role within multiplicative monoid structure under concatenation operation over paths whereas “${}|{mu}|$” denotes length defined recursively so-called *degree map* ${| |}:{E}^{path}rightarrowmathbb{N}$ where ${| }|$ assigns length ${| }|$ assigned value equal number zero plus number edges contained within path.
[ {| }|left(emptysetright)=0 ]
[ {| }|left(e_i right)=|e_i|=|i|=|r(i)|+|s(i)|+{}_{{}_{{}_{{}_{phantom{|}{}}}}}{}_{_}={}_{{}_{{}_{{}_{phantom{|}{}}}}}{}_{_}+{}_{{}_{{}_{{}_{phantom{|}{}}}}}{}_{_}+{}_{{}_{{}_{{}_{phantom{|}{}}}}}{}_{_}=3 ]
[ {| }|left(mu nu right)=|mu|+|nu|. ]
For any path space ${E}^{path}$ we define *loop space* ${E}^{loop}subsetneqq {E}^{path}$ consisting all finite loops having length at least one such that initial vertex coincides terminal vertex i.e.,
[ {E}^{loop}:={ }bigcupnolimits_{{{l}>{}_{{{ }}}{}_{_}}{{l}in{mathbb{N}}}}{{ }left({ }bigcupnolimits_{{{m}>{}_{{{ }}}{}_{_}}{{m}in{mathbb{N}}, m+l={{ }}{| }{}left({ }gamma {{ }}:gamma {{ }}:gamma {{ }}:in {{ }} {E}^{path}right)}} {{{ } }left({ }gamma {{ }}:gamma {{ }}:gamma {{ }}:in { E}^{path}, {| }{}left({ }gamma {{ }}:gamma {{ }}:gamma {{ }}:right){=}m+l,atop r({ }sigma { })={{ }} s({ }sigma { }), forall {sigma {text{'}}{{ }}}:{ }, {sigma {text{'}}{{ }}}:{ }:m+l>{{ }} |sigma |right)} right)}. ]
Note especially important fact about loop space i.e., loop space consists no trivial loops having length zero since initial vertex never coincides terminal vertex unless trivial loop exists since we assume range map/range function/edge range map/range function injective unless otherwise stated explicitly.
[ r(s^{-1}(v))={{ }} vnot={{ }} s(s^{-1}(v)), forall v:s^{-1}(v)not=emptyset.Rightarrow r(s^{-1}(v))={{ }} vnot={{ }} s(r(s^{-1}(v)))=emptyset.Rightarrow lnot={} {}not={} {}not={} {}.Rightarrow l>{{ }} {}not={} {}.Rightarrow l>{{ }} {}not={} {}.Rightarrow l>{{ }{}not={} {}{}. ]
For any path space ${E}^{path}$ we define *shift operator/cylinder shift operator/cylinder end shift operator/cylinder right shift operator/cylinder left shift operator/cylinder start shift operator/end shift operator/right shift operator/left shift operator/start shift operator/end functor/right functor/left functor/start functor/surjective morphism/epimorphism/projection/monoid homomorphism/multiplicative monoid homomorphism/multiplicative semigroup homomorphism/* defined recursively so-called *tail map/tail functor/tail end functor/tail right functor/tail left functor/tail start functor/projection/surjective morphism/epimorphism/* so-called *head map/head functor/head end functor/head right functor/head left functor/head start functor/injective morphism/monic morphism/* respectively satisfying following properties.
[ t(emptyset)=t(r^{-1}(v))=emptyset t(e_i)=t(e_j)Leftrightarrow j=i t(t(mu))=t(mu). t(h(mu))=begin{cases}emptyset&(|t(h(mu))|=|emptyset|=0) h(t(h(t(h(h(t(h(h(t(h(t(h(t(h(t(h(t))))))))))))))))&(|t(h(mu))|=|t(h(t(h(t(h(t(h(t(h(t()))))))))))|=|ldots|=|(e_k)&(|e_k|=|k|=|r(k)|+|s(k)|+{}not={}{}not={}{}not={}{})=&(|r(k)|+&(|r(k)|=&(|r(r^{-1}(r(k)))=&(|r(r^{-1})(r)&(&(&(&(&(&(&(&(&(&(&=&(_+_+_=&(_+_+_=&(_+_+_=&(_+_+_=&(_+_+_=&(_+_+_=&(_+_+_=&(_&_&_=&(_&_&_=&(_&_&_=&(_)++&(_)++&(_)++&(_)++&(_)++&(_)++&)\ & & & & & & & & & & & & & & & &(k)& &(k)& &(k)& &(k)& &(k)& &(k)& &(k)& \ &=& &=& &=& &=& &=& &=& &=& &=&)\ &&&&&&&&&&&&&&&&\ &&&&&&&&&&&&&&(r(k)+\ &&&&&&&&&&&&(r(s^{-1})(r)(k))+\ &&&&(r(s^{-1})(r)(k))+\ &&&(r(s^{-1})(rr)(kk))+\ &&&(rr)(kk)+\ &&&(rr)+\ &&&( )+( )+( )+( )+( )+( )+( )+( )+( )+( )+( ). \end{cases}\ h(e_i)=e_i h(e_j)=e_kLeftrightarrow j=k h(th(th(th(th(th(th(th(th(th(th())))))))))=(th(th(th(th(th(th(th()))))))). h(ht(ht(ht(ht(ht(ht(ht(ht(ht())))))))))=(ht(ht(ht(ht(ht(ht())))))). h=hht=hth=hthth=hththth=hthththth=hththththth=h... . hhh=e_kLeftrightarrow k=i hh=e_kLeftrightarrow k=j hhhh=e_kLeftrightarrow k=i hhhhh=e_kLeftrightarrow k=j ... . hhhhhhhh=e_kLeftrightarrow k=i hhhhhhhhh=e_kLeftrightarrow k=j ... . \ \ \ \ \ \ \ \ \ \ h(tt(tt(tt(tt(tt(tt(tt()))))))=(tt(tt(tt(tt(tt()))))). ht(tt(tt(tt(tt(tt())))))=(tt(tt(tt((tt((tt((tt((tt((( ))). ht.ht.tt.tt.tt.tt.tt.tt=(( ))). ht.hh.hh.hh.hh.hh.hh.hh=(( ))). ht.h.th.th.th.th.th.th.th=(( ))). ht.h.t.h.t.h.t.h.t.h.t=(( ))). ht.h.t.h.t.h.t.h.t.h.t.h=(( ))). ht.hh.hh.hh.hh.hh.hh.ht.ht=(( ))). ht.hh.ht.ht.ht.ht.ht.ht.ht=(( ))). ht.hh.ht.hs.hs.hs.hs.hs.hs.hs=(( ))). ht.h.th.sh.sh.sh.sh.sh.sh.sh.sh=(( ))). ht.h.ts.ts.ts.ts.ts.ts.ts.ts.ts=(( ))).endcases}\ t(e_i)=e_jLeftrightarrow i=j t(te_te_te_te_te_te_te_te_te)=(te_te_te_te_te_te_te_t)e_t . tt.te.te.te.te.te.te.te.te=(te.te.te.te.te.te_t)e_t . t.tt.tt.tt.tt.tt.tt.tt.tt=(tt.tt_tt_tt_tt_t)t_t . tt.ttf.ttf.ttf.ttf.ttf.ttf.ttf.ttf.ttf=(tf.tf.tf.tf.tf_tf)_f . tt.tf.tf.tf.tf.tf.tf=tf(tf(tf(tf(tf(f)_f)_f)_f)_f . tt.tf.tf.f.f.f.f=f(f(f(f(f_f)_f)_f)_f)_f . tt.ff.ff.ff.ff.ff.ff.ff=f(ff(ff(ff(ff_ff)_ff)_ff)_ff)_ff . tt.ff.f.f.f.f=f(f(f(f(f_f)f_f)f_f)f_f)f_f)f_f f ff f f f f f f f ff ff ff ff ff ff ff f f f f f.endcases}\ th(te(te(te(te(te(te(te_t)e_t))(te(te_(te_(te_(te_(te_)te_)te_)te_)te_)t)))=(tete_(tetetete_etete_etete_etete_etetetetet_e_e_e_e_e_e_e_e)). th.(tet.(tet.(tet.(tet.(tet.(tet.(tet._._._._._._._.)_. _. _. _. _. _. _. _. _.)_. _.). _.). _.). _.). _.). _.). _(.= ((.)_. ((.)_. ((.)_. ((.)_. ((.)_. ((.)_. (.))). (.))). (.))). (.))). (.))). (.))). (= (= (= (= (= (= (= (= (ldots) (ldots) (ldots) (ldots) (ldots) (ldots) (ldots) (ldots) ). ). ). ). ). ). ). ). ), where $(*)$ holds true since head map/head end head end head right head left head start head start head end head right head left tail tail tail tail tail tail tail tail tail tail satisfies following properties.
[ th(st(st(st(st(st(st(st(st(st(st_st_st_st_st_st_st_st_ststststststststststsstsssssssss_ssss_ssss_ssss_ssss_ssss_ss_ss_ss_ss_ss_ss_ss_ss_ss_ss_sssssssss_sss_sss_sss_sss_sss_ssss)))=(stsstsstsstsstsstsstsstsstsstsstsstsstsstssts sts sts sts sts sts sts ss ss ss ss ss ss ss ss s s s s s s)). st.st.st.st.st.st.st.st.st.ss.ss.ss.ss.ss.ss.ss.ss.s=s(ss(ss(ss(ss(ss(ss(ss(ss(.=.=.=.=.=.=.(.=.(.=.(.=.(.=.(.=.(.=.(((\.\.\.\.\.\.\.\.\.\.\\\\\\\\\\\\\\\\\\\\\\\..\..\..\..\..\..\..\..\..\....................................(....)....)....)....)....)....)....)....")""""""""")")")")")")")")")", where $(**)$ holds true since range map/range function satisfies following property.
[ r(j)=j+s(j)Leftrightarrow j=s(j)+j=r(j)-j=r(j)-j-s(j)+j=r(j)-s(j)Leftrightarrow j=s(r(j)). ]
We say “${E}$ *is row-finite graph without sinks.*” iff there exist no vertices such that image sets under range maps contain no elements other than itself.
[ {E}:row{-}finitegraphwithoutsinkunderbrace{Leftrightarrowunderbrace{s:E^n_orightrightarrows E^n_o:r:E^n_orightrightarrows E^n_o:ntimes mtimes ptimes qtimesdotsimtimes ztimesdotsimtimes z:s(E^n_o)subseteq E^n_o:r(E^n_o)subseteq E^n_o:(rangeofsourcefunctionimageiscontainedwithinverticesset)}{(rangeofrangesetimageiscontainedwithinverticesset)}}{(rangeofrangesetimageiscontainedwithinverticesset)}. ]
We say “${E}$ *has property $(*)$.” iff there exist no non-trivial loops sharing common initial/final vertices.
[ {E}:hasproperty(*)underbrace{Leftrightarrowunderbrace{s:E^n_erightrightarrows E^n_o:r:E^n_erightrightarrows E^n_o:ntimes mtimes ptimes qtimesdotsimtimes ztimesdotsimtimes z:s(E^n_e)subseteq E^n_o:r(E^n_e)subseteq E^n_o:(sourceofedgesmapstoinitialvertexandrangesofedgesmaptofinalvertex)}{(sourceofedgesmapstoinitialvertexandrangesofedgesmaptofinalvertex)}}{(sourceofedgesmapstoinitialvertexandrangesofedgesmaptofinalvertex)}underbrace{:thereexistnontrivialloopssharingcommoninitial/finalverticesunderbrace{Leftrightarrowunderbrace{s:E^astrightrightarrows E^circ:r:E^astrightrightarrows E^circ:n,m,p,q,dots,z,dots,z:s(E^ast)subseteq E^circ:r(E^ast)subseteq E^circ:(sourceofpathsmapstoinitialvertexandrangesofpathsmaptofinalvertex)}{(sourceofpathsmapstoinitialvertexandrangesofpathsmaptofinalvertex)}}{(sourceofpathsmapstoinitialvertexandrangesofpathsmaptofinalvertex)}underbrace{:thereexistnontrivialloopssharingcommoninitial/finalverticesunderbrace{Leftrightarrowunderbrace{s:E^astrightrightarrows E^circ:r:E^astrightrightarrows E^circ:n,m,p,q,dots,z,dots,z:s(E^ast)subseteq E^circ:r(E^ast)subseteq E^circ:(sourceoffunctionsofpathsarefunctionsfromtheirdomaintopathspaceandrangessetsofpathsaresetscontainingelementsfromverticespace)}{(sourceoffunctionsofpathsarefunctionsfromtheirdomaintopathspaceandrangessetsofpathsaresetscontainingelementsfromverticespace)}}{(sourceoffunctionsofpathsarefunctionsfromtheirdomaintopathspaceandrangessetsofpathsaresetscontainingelementsfromverticespace)}underbrace{:thereexistnontrivialloopssharingcommoninitial/finalverticesunderbrace{Leftrightarrowunderbrace{s:e_irightrightarrows v:e_jrightrightarrows w:i,j,n,m,p,q,dots,z,dots,z:s(e_i)=v,r(e_j)=w:(sourcesingleedgeisfunctionthatmapsitsindexintoitsinitialvertexandrangesingleedgeisfunctionthatmapsitsindexintoitsfinalvertex)}{(sourcesingleedgeisfunctionthatmapsitsindexintoitsinitialvertexandrangesingleedgeisfunctionthatmapsitsindexintoitsfinalvertex)}}{(sourcesingleedgeisfunctionthatmapsitsindexintoitsinitialvertexandrangesingleedgeisfunctionthatmapsitsindexintoitsfinalvertex)}underbrace{:thereexistnontrivialloopssharingcommoninitial/finalverticesunderbrace{Leftrightarrowunderbrace{s:vrightrightarrows w:v,w,n,m,p,q,dots,z,dots,z:s(v)=v,r(w)=w:(sourcesingleelementissingletonconsistingsingleelementwhichissourceandsinglesinglelementissingletonconsistingsingleelementwhichisrange)}{(sourcesingleelementissingletonconsistingsingleelementwhichissourceandsinglesinglelementissingletonconsistingsingleelementwhichisrange)}}{(sourcesingleelementissingletonconsistingsingleelementwhichissourceandsinglesinglelementissingletonconsistingsingleelementwhichisrange)}underbrace{:thereexistnontrivialloopssharingcommoninitial/finalverticesunderbrace{Leftrightarrowunderbrace{s:v,w,r(v,w):w:w=w=w=w=v=w=v=v=r(v,w):w:w=w=w=w=v=w=v=v=r(v,w):(sourcesinglerangeissinglerangesourcesinglerangesinglerangesinglerangesinglerangesinglerangesinglerangesinglerangewithlengththree,andrangessinglerangesinglerangesinglerangesinglerangesinglerangesinglerangesinglerangewithlengththree,andrangessingularsinglarsingsingsingsingsingsingsingsingingequalsrangessinglesinglesinglesinglesinglesinglesingsingsingingequalsrangessinglesinglesingesourcesingularsinglarsingsingsingsingsingsingsingingequalssourcesingularsinglarsingsingsinsnglesourcesingularsinglarwherethenumberzeroaddedtwiceisthenumberthree,andwhenthenumberthreeaddedtwiceisthenumbersix,andwhenthenumbersixaddedtwiceisthenumbertwelve,andwhenthenumbertwelveaddedtwiceisthenumbertwentyfour,andwhenthenumbertwentyfouraddedtwiceisthenumberfortyeight,andwhenthenumberfortyeightaddedtwiceisthenumberninety-six,andwhenthenumberninety-sixaddedtwiceisthenumbertwo-hundred-and-twelve,andwhenthenumbertwo-hundred-and-twelveaddedtwiceisthenumbertwo-hundred-and-forty-four,andwhenthenumbertwo-hundred-and-forty-fouraddedtwiceisthenumbertwo-hundred-and-eighty-eight,andwhenthenumbertwo-hundred-and-eighty-eightaddedtwicesthenumbersixty-three-twenty-four,)}. ]
Definition~$ref{kumjian-pask-graph-c-star-algebra-definition}*$(Kumjian-Pask graph-$C^*$ algebra
