# Questions tagged [baer-rings]

The baer-rings tag has no usage guidance.

17
questions

**2**

votes

**0**answers

83 views

### Algebraic language of fundamental results in operator algebras

Ignoring topological structures of von Neumann algebras, the larger category of Baer $*$-rings was emerged. In the unique text written by Sterling K. Berberian (1), some concepts and results in von ...

**1**

vote

**0**answers

38 views

### Relation between left projections

Let $A$ be a Baer *-ring. Let $x$ be in $A$, $L(x)$ is the left projection of $x$ that is the smallest projection with $L(x)x=x$.
Q. Let $p,q$ are projections in $A$ with $p\leq q$. For a given ...

**1**

vote

**0**answers

35 views

### The statue of a sequence of finite projections

Let $A$ be a Baer $*$-ring. Let $\{p_n\}$ be a sequence of finite projections in $A$. True or false?
Suppose that there is no $N$ with $p_n=p_{n+1}$ for $n\geq N$. We have then $\inf_{1\leq n\leq ...

**1**

vote

**0**answers

34 views

### something concerning finite projections

Let $A$ be a Baer *-ring. Let $x$ be an isometry (meaning $x^*x=1$ where $1$ is the unit of $A$).
Let $e$ be a finite projection in $A$ such that $ex^ne=ex^n$ for every $n\geq0$.
Q. Can we say that ...

**1**

vote

**2**answers

109 views

### Upper triangular $2\times2$-matrices over a Baer *-ring

Let $A$ be a Baer $*$-ring. Let us denote $B$ by the space of all upper triangular matrices
$\left(\begin{array}{cc}
a_1& a_2 \\
0 & a_4
\end{array}\right)$ where $a_i$'s are in $A$. Is $B$ ...

**2**

votes

**1**answer

71 views

### Strongly finite projections in $*$-rings

Let $A$ be a $*$-ring. Let us have some points:
i) We recall that a projection $p$ is a self-adjoint idempotent that is $p=p^*=p^2$.
ii) On the set of projections, we write $p\leq q$ if $pq=p$.
iii)...

**2**

votes

**0**answers

66 views

### Transmission of finite projections

Let $A$ be a Baer*-ring. Let us denote $L(x)$ by the left projection of $x$ (the smallest projection with $L(x)x=x$).
Let $p$ be a finite projection in $A$. Is $L(xp)$ a finite projection for every $...

**2**

votes

**0**answers

141 views

### Algebraic version of unilateral shift

It was confirmed that Wold-type decomposition can be extended from von Neumann algebras to Baer*-rings (see this paper). By algebraic tools the notion of unilateral shifts is successfully transmitted ...

**1**

vote

**0**answers

109 views

### shifts in Baer*-rings

Let $A$ be a Baer*-ring with the unit $1$. An important point that holds in every Baer*-ring is this: for every element $y\in A$ there is an smallest projection $e_y\in A$, called the left projection ...

**2**

votes

**0**answers

106 views

### When does a projection commute with an element?

Let $A$ be a Baer*-ring and $a$ be in $A$. Assume that for a projection $p\in A$,
the following system of equations hold:
\begin{array}{ll}
a^na^{*n}p=p , & n\geq0\\
a^{*n}a^np=p , &...

**2**

votes

**1**answer

101 views

### Is $M_n(Z_p)$ a Baer*-ring?

Let $p$ be a prime number and $n$ be a natural number. Does $M_n({\mathbf{Z}}_p)$, the ring of all $n\times n$ matrices over the field ${\mathbf{Z}}_p$ form a Baer*-ring?
If not, what about for ...

**1**

vote

**0**answers

71 views

### Does the inequality $x^2\leq x$ hold in Baer*-ring for $0 \leq x \leq 1$?

Let $A$ be a Baer*-ring and $x$ be a positive element with $0\leq x\leq1$ where $1$ is just the unit of $A$. Can we conclude $x^2\leq x$?

**1**

vote

**1**answer

154 views

### A Baer *-ring which is not embedded into $B(H)$

Assume $A$ is a complex $*$-algebra which is also a Baer*-ring.
Q. Can we concluded that there exists a Hilbert space $H$ such that $A$ is embedded in $B(H)$ as a Baer*-ring? What about when $A$ ...

**1**

vote

**0**answers

58 views

### square root of contractions in Baer *-rings

Let $A$ be a unital Baer*-ring. We say that $a$ is a contraction if $aa^*\leq1$ and $a^*a\leq1$.
Q1) Assume $a$ is a contraction. Has the positive element $1-aa^*$ any square root?
(if yes, seems $1-...

**1**

vote

**3**answers

182 views

### Some non-trivial Baer *-rings

A Baer *-ring is an *-algebra whose lattice of projections is complete. I know two well-handed kinds of these structures:
1- W*-algebras (abstract case of von Neumann algebras).
2- The inverse ...

**1**

vote

**1**answer

78 views

### Two points concerning Baer *-rings

Let $A$ be a unital Baer *-ring.
1- Assume that $\{p_i\}$ is a family of projections in $A$. Let $x$ be an isometry in A (I mean $x^*x=1$ where $1$ is the unit of $A$). True or false: $\inf (...

**1**

vote

**1**answer

166 views

### Normal reduced rings

It is known that a reduced commutative unitary Baer ring $A$ is normal iff for every prime ideal $\mathfrak p$ of $T(A)$ (here $T(A)$ is the total quotient of $A$) one has $A/(\mathfrak p\cap A)$ is ...