Conditions for Primitivity of unital amalgamated full free products of finite dimensional C*-algebras

We consider amalgamated unital full free products of the form $A_1*_DA_2$, where $A_1, A_2$ and $D$ are finite dimensional C*-algebras and there are faithful traces on $A_1$ and $A_2$ whose restrictions to $D$ agree. We provide several conditions on the matrices of partial multiplicities of the inclusions $D\hookrightarrow A_1$ and $D\hookrightarrow A_2$ that guarantee that the C*-algebra $A_1*_DA_2$ is primitive. If the ranks of the matrices of partial multiplicities are one, we prove that the algebra $A_1*_DA_2$ is primitive if and only if it has a trivial center.Comment: 40 pages. No essential changes in second submission. Some misspelled words were corrected and format was change

Rich Words and Balanced Words

This thesis is mostly focused on palindromes. Palindromes have been studied extensively, in recent years, in the field of combinatorics on words.Our main focus is on rich words, also known as full words. These are words which have maximum number of distinct palindromes as factors.We shed some more light on these words and investigate certain restricted problems. Finite rich words are known to be extendable to infinite rich words. We study more closely how many different ways, and in which situations, rich words can be extended so that they remain rich.The defect of a ord is defined to be the number of palindromes the word is lacking.We will generalize the definition of defect with respect to extending the word to be infinite.The number of rich words, on an alphabet of size $n$, is given an upper and a lower bound. Hof, Knill and Simon presented (Commun. Math. Phys. 174, 1995) a well-known question whether all palindromic subshifts which are enerated by primitive substitutions arise from substitutions which are in class P. Over the years, this question has transformed a bit and is nowadays called the class P conjecture. The main point of the conjecture is to attempt to explain how an infinite word can contain infinitely many palindromes.We will prove a partial result of the conjecture. Rich square-free words are known to be finite (Pelantov\'a and Sarosta, Discrete Math. 313, 2013). We will give another proof for that result. Since they are finite, there exists a longest such word on an $n$-ary alphabet.We give an upper and a lower bound for the length of that word. We study also balanced words. Oliver Jenkinson proved (Discrete Math., Alg. and Appl. 1(4), 2009) that if we take the partial sum of the lexicographically ordered orbit of a binary word, then the balanced word gives the least partial sum. The balanced word also gives the largest product. We will show that, at the other extreme, there are the words of the form $0^{q-p}1^p$ ($p$ and $q$ are integers with $1\leq p<q$), which we call the most unbalanced words. They give the greatest partial sum and the smallest product.Tässä väitöskirjassa käsitellään pääasiassa palindromeja. Palindromeja on tutkittu viime vuosina runsaasti sanojen kombinatoriikassa.Suurin kiinnostuksen kohde tässä tutkielmassa on rikkaissa sanoissa. Nämä ovat sanoja joissa on maksimaalinen määrä erilaisia palindromeja tekijöinä.Näitä sanoja tutkitaan monesta eri näkökulmasta. Äärellisiä rikkaita sanoja voidaan tunnetusti jatkaa äärettömiksi rikkaiksi sanoiksi.Työssä tutkitaan tarkemmin sitä, miten monella tavalla ja missä eri tilanteissa rikkaita sanoja voidaan jatkaa siten, että ne pysyvät rikkaina.Sanan vajauksella tarkoitetaan puuttuvien palindromien lukumäärää.Vajauksen käsite yleistetään tapaukseen, jossa sanaa on jatkettava äärettömäksi sanaksi.Rikkaiden sanojen lukumäärälle annetaan myös ylä- ja alaraja. Hof, Knill ja Simon esittivät kysymyksen (Commun. Math. Phys. 174, 1995), saadaanko kaikki äärettömät sanat joissa on ääretön määrä palindromeja tekijöinä ja jotka ovat primitiivisen morfismin generoimia, morfismeista jotka kuuluvat luokkaan P. Nykyään tätä ongelmaa kutsutaan luokan P konjektuuriksi ja sen tarkoitus on saada selitys sille,millä tavalla äärettömässä sanassa voi olla tekijöinä äärettömän monta palindromia. Osittainen tulos tästä konjektuurista todistetaan. Rikkaiden neliövapaiden sanojen tiedetään olevan äärellisiä (Pelantov\'a ja Starosta, Discrete Math. 313, 2013). Tälle tulokselle annetaan uudenlainen todistus.Koska kyseiset sanat ovat äärellisiä, voidaan selvittää mikä niistä on pisin.Ylä- ja alaraja annetaan tällaisen pisimmän sanan pituudelle. Työssä tutkitaan myös tasapainotettuja sanoja.Tasapainotetut sanat antavat pienimmän osittaissumman binäärisille sanoille (Jenkinson, Discrete Math., Alg. and Appl. 1(4), 2009).Lisäksi ne antavat suurimman tulon.Muotoa $0^{q-p}1^p$ ($p$ ja $q$ ovat kokonaislukuja joille $1\leq p<q$) olevien sanojen todistetaan vastaavasti antavan suurimman osittaissumman ja pienimmän tulon.Ne muodostavat täten toisen ääripään tasapainotetuille sanoille, ja asettavat kaikki muut sanat näiden väliin.Siirretty Doriast

Extremal words in morphic subshifts

Given an infinite word X over an alphabet A a letter b occurring in X, and a total order \sigma on A, we call the smallest word with respect to \sigma starting with b in the shift orbit closure of X an extremal word of X. In this paper we consider the extremal words of morphic words. If X = g(f^{\omega}(a)) for some morphisms f and g, we give two simple conditions on f and g that guarantees that all extremal words are morphic. This happens, in particular, when X is a primitive morphic or a binary pure morphic word. Our techniques provide characterizations of the extremal words of the Period-doubling word and the Chacon word and give a new proof of the form of the lexicographically least word in the shift orbit closure of the Rudin-Shapiro word.Comment: Replaces a previous version entitled "Extremal words in the shift orbit closure of a morphic sequence" with an added result on primitive morphic sequences. Submitte

Abelian Primitive Words

We investigate Abelian primitive words, which are words that are not Abelian powers. We show that unlike classical primitive words, the set of Abelian primitive words is not context-free. We can determine whether a word is Abelian primitive in linear time. Also different from classical primitive words, we find that a word may have more than one Abelian root. We also consider enumeration problems and the relation to the theory of codes